Question

How do you use the fundamental trigonometric identities to determine the simplified form of the expression?

Answer 1

"The fundamental trigonometric identities" are the basic identities:

•The reciprocal identities
•The pythagorean identities
•The quotient identities

They are all shown in the following image:

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When it comes down to simplifying with these identities, we must use combinations of these identities to reduce a much more complex expression to its simplest form.

Here are a few examples I have prepared:

a) Simplify: `tanx/cscx xx secx`

Apply the quotient identity `tantheta = sintheta/costheta` and the reciprocal identities `csctheta = 1/sintheta` and `sectheta = 1/costheta`.

`=(sinx/cosx)/(1/sinx) xx 1/cosx`

`=sinx/cosx xx sinx/1 xx 1/cosx`

`=sin^2x/cos^2x`

Reapplying the quotient identity, in reverse form:

`=tan^2x`

b) Simplify: `(cscbeta - sin beta)/cscbeta`

Apply the reciprocal identity `cscbeta = 1/sinbeta`:

`=(1/sinbeta - sin beta)/(1/sinbeta)`

Put the denominator on a common denominator:

`=(1/sinbeta - sin^2beta/sinbeta)/(1/sinbeta)`

Rearrange the pythagorean identity `cos^2theta + sin^2theta = 1`, solving for `cos^2theta`:

`cos^2theta = 1 - sin^2theta`

`=(cos^2beta/sinbeta)/(1/sinbeta)`

`=cos^2beta/sinbeta xx sin beta/1`

`=cos^2beta`

c) Simplify: `sinx/cosx + cosx/(1 + sinx)`:

Once again, put on a common denominator:

`=(sinx(1 + sinx))/(cosx(1 + sinx)) + (cosx(cosx))/(cosx(1 + sinx))`

Multiply out:

`=(sinx + sin^2x + cos^2x)/(cosx(1 + sinx))`

Applying the pythagorean identity `cos^2x + sin^2x = 1`:

`=(sinx + 1)/(cosx(1 + sinx))`

Cancelling out the `sinx + 1` since it appears both in the numerator and in the denominator.

`=cancel(sinx + 1)/(cosx(cancel(sinx + 1))`

`=1/cosx`

Applying the reciprocal identity `1/costheta = sectheta`

`=secx`

Finally, on a last note, I know that here in Canada, British Columbia more specifically, these identities are given on a formula sheet, but I don't know what it's like elsewhere. In any event, many students, me included, memorize these identities because they're that important to mathematics. I would highly recommend memorization.

Practice exercises:

Simplify the following expressions:

a) `cosalpha + tan alphasinalpha`

b) `cscx/sinx - cotx/tanx`

c) `sin^4theta - cos^4theta`

d) `(tan beta + cot beta)/csc^2beta`

Hopefully this helps, and good luck!